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In this section we will proceed informally, neither proving our claims nor even carefully defining our terms. Nevertheless, as you will see in the course of reading this book, everything we say here is absolutely correct. We proceed in this way to show in advance what our main goals are, and hence to motivate our development.

We begin by defining the objects we will be studying.

We now apply our general theory to the case of symmetric functions. We let be an arbitrary field and set (,⋯, ), the field of rational functions in the variables ,⋯, . Then the symmetric group acts on by permuting ,⋯,

In this section we deal with a number of questions about polynomials in [] related to factorization and irreducibility.

We now wish to further investigate questions related to separability and inseparability of algebraic extensions. Recall from Corollary 3.2.3 that every algebraic extension in characteristic 0 is separable, so in this case there is nothing more to be said. char()=>0.